Few-cycle intense femtosecond pulses are an important and an enabling tool in time-resolved spectroscopy measurements and particularly in attosecond physics. However, despite essential progress in the development of chirped pulse amplifiers (CPAs), in practice, the shortest pulses generated from such a source have only reached 15 femtoseconds (fs). This is to be traced back to a fundamental limitation, namely gain narrowing in the laser crystal that results in reduced spectral bandwidths, and consequently in increased pulse durations. Hollow fiber compression (HFC) allows to overcome this limitation by broadening the spectrum of the pulses generated with femtosecond amplifiers. The spectrally broadened pulses are subsequently compressed in a negative dispersion delay line.
HFC relies on the nonlinear propagation of intense femtosecond pulses in gas-filled capillary tubes.
In FIG. 1, a schematic representation of an arrangement of a standard HFC setup is shown; in FIG. 2, a more detailed illustration of such a setup, partly in section, is given.
According to these FIGS. 1 and 2, a laser beam 1 generated in a laser amplifier 2 is coupled into a fiber waveguide 3, i.e. into a hollow fiber 3A (FIG. 2) placed in an air-tight hollow fiber chamber 3B via a folding mirror 4 and a focusing mirror 5. At the output of the chamber 3B, the beam is re-collimated with a mirror 6 and directed into a compressor 7, for instance comprised by a negative-dispersion delay line 8.
Furthermore, FIG. 3 shows a cross-section of a hollow fiber waveguide 3A consisting of a glass cladding 9 and hollow core 10 and having an inner diameter equal to 2 a.
Self-phase-modulation in a hollow fiber waveguide leads to spectral broadening while the spatial filtering properties of the waveguide ensure a transversally homogeneous spectral broadening and result in an excellent beam profile. Initially devised for sub-mJ pulses (see for instance M. Nisoli et al., “A novel-high energy pulse compression system: generation of multigigawatt sub-5-fs pulses”, Appl. Phys. Lett. B 65 (1997): p. 189-196), the method enabled the generation of sub-TW few-cycle pulses at KHz repetition frequencies from table-top systems. Subsequently, CPAs equipped with HFCs became the working horse of many femto- and atto-second physics laboratories around the world. However, several applications including high-photon yield high order harmonic generation, the generation of high-photon energy harmonic radiation, or the investigation of relativistic laser field-matter interactions call for few-cycle driving laser pulses with energies in excess of 1 mJ. With CPAs supplying multi-mJ pulses with durations in the range of 30-50 fs widely available, energy up-scaling of pulse compressors gained paramount importance for the development of laser sources for strong-field physics. However, energy up-scaling of HFC is not trivial since several phenomena affect the performance of HFCs when the pulse energy is increased beyond 1 mJ, as for instance ionization of the propagation medium, damage of the fiber entrance, self-phase modulation and self-focusing in front of the fiber.
Light propagation in a hollow fiber waveguide can be described by decomposing the optical field in discrete spatial modes with transversal mode profiles given byVjp=Jj(ujpr/a)where
Jj with j=0,1, . . . is the Bessel function of order j;
ujp with p=1,2, . . . is the pth root of the equation Jj(ujp)=0;
r is the radial coordinate (in a cylindrical coordinates system having its symmetry axis collinear with the symmetry axis of the hollow fiber); and
a is the fiber radius (see FIG. 3).
The longitudinally polarized (LP) modes having a spatial profile Vjp are dubbed the LPjp modes. The complex propagation constant kjp of the mode jp is given bykjp(ω)=βjp(ω)+iαjp(ω)with
ω being the angular frequency of the laser light; and
βjp, αjp being the real, respectively imaginary parts of the propagation constant.
The modes thus have different attenuation constants αjp(ω)=2.814ujp/a3[λ/(2π)]2,
with
λ=wavelength of the laser light;
and this accounts for the spatial filtering properties of the fiber.
The mode LP01 exhibits the smallest propagation losses and is referred to as fundamental or lowest-order mode. This mode LP01 has a profile given by a first order Bessel function (compare also FIG. 4, full line graph) that has zero-transitions at the fiber walls (i.e. for r=+a and r=−a) and is a very close approximation of a Gaussian mode, and it can be efficiently focused by means of standard optics. Modes of increasingly higher order have increasingly complex transversal profiles that deviate significantly from a Gaussian profile and result in a poor focusability of the beam. The first order mode LP02 (compare FIG. 4, graph in dotted line), for instance, is given by a first order Bessel function that has zero-transitions at the fiber walls (for r=a and r=−a) and additionally two minima within the fiber core; this mode has the second-lowest propagation losses. Propagation through the fiber discriminates thus among the modes, and after a sufficiently long propagation distance the energy will be substantially contained in the (well-focusable) fundamental mode LP01 since higher-order modes experience much stronger attenuation.
In the presence of a nonlinear effect occurring during propagation in the waveguide (either the Kerr-effect or ionization), energy is coupled from the fundamental mode LP01 to higher-order modes (cf. G. Tempea and T. Brabec, “Theory of self-focusing in a hollow wa-waveguide”, Opt. Lett. 23 (10) (1998): p. 762-764) which might lead to a degradation of the beam profile. Given the extremely high losses and low coupling coefficients of modes with j>2, it is sufficient to consider only coupling to the mode LP02 for the analysis of this phenomenon. The difference between the real parts β01(ω) and β02(ω) of the propagation constants kjp of the modes LP01 and LP02 and between their first order derivatives (with respect to the angular frequency ω) leads to phase mismatch (i.e. the phase of the two modes changes at different rates during propagation), or to group velocity mismatch, respectively (the two modes propagate with different velocities) between the modes. These phenomena can be quantified by means of the following parameters: the phase-mismatch length Lp12 can be described asLp12=λ0/[4πa2(u022−u012)], with
λ0=carrier wavelength of the pulse to be compressed
(Lp12 being equal to 2×10−6×a2 for a center wavelength λ0 of 800 nm) and is the propagation length required for a phase difference of π to build up between the phases of the modes LP01 and LP02; the group-velocity mismatch length Lv12 can be described asLv12=2τFWHMcLp12/λ0,with
τFWHM=half maximum pulse duration; and
c=speed of light in vacuum,
and is the propagation length required for a delay equal to the full width at half maximum pulse duration τFWHM to build up between the modes LP01 and LP02.
The physical meaning of these two quantities is the following: energy is coupled from the mode LP01 and LP02 as long as the phase difference between the two modes is <π, i.e. for propagation length l<Lp12. Subsequently, energy is coupling back to the fundamental mode LP01 for propagation lengths LP12<l<2 LP12 and this process of periodic energy transfer between the modes LP01 and LP02 is repeated as long as there is still significant temporal overlapping between the pulses propagating with the two spatial modes, i.e. for lengths l<Lv12.
Corresponding to the three phenomena governing mode-coupled propagation (phase-mismatch, group velocity mismatch and mode-dependent losses), three regimes A, B, C of propagation can be identified:
(A) The Phase-mismatched Periodic Mode-coupling Propagation:                The group velocity mismatch has negligible effects (i.e. pulses having spatial modes LP01 and LP02 propagate almost synchronously); the energy transfer between LP01 and LP02 is periodic due to phase-mismatch.        
(B) Reduced-coupling Propagation:                The group velocity mismatch reduces the temporal overlapping between the pulses significantly; furthermore, pulse stretching and losses result in reduced nonlinearity and, therewith, in reduced mode coupling; the power of the mode LP02 is already reduced due to the large propagation losses.        
(C) De-coupled Propagation:                The delay between the pulses traveling in the two propagation modes becomes comparable with the duration of the pulses resulting in negligible temporal overlap; pulse broadening and losses reduce nonlinear mode-coupling to negligible levels. The two modes propagate substantially independently, exhibiting the mode-specific propagation losses; after sufficiently long propagation distances, the energy contained in the LP02 mode will be negligible in comparison to the energy contained in the fundamental mode.        
Hollow fiber compression schemes proposed so far rely on the loss-related mode discrimination mechanism; the fiber length is chosen such that the transmittance of the mode LP02 is negligible as compared to the transmittance of the mode LP01 (see U.S. Pat. No. 5,956,173 A; and M. Nisoli et al., “A novel-high energy pulse compression system: generation of multigigawatt sub-5-fs pulses”, Appl. Phys. Lett. B 65 (1997): p. 189-196; in particular p. 190, FIG. 1b and the corresponding discussion at the end of section 1). This method is well applicable for the compression of pulses with energies of approximately 1 mJ and slightly beyond, where fibers with diameters up to 200 μm-300 μm can be used. Compression of more energetic pulses (with energies well beyond 1 mJ) calls for the use of fibers with larger diameters in order to avoid excessive nonlinearities and/or damage of the fiber entrance. However, the loss-related mode discrimination of the fiber decreases rapidly with increasing fiber radius a according to the formula:(α01(ω)−α02(ω))=2.814(u01−u02)/a3[λ/(2π)]2 
Consequently, very long fibers need to be used in order to achieve proper mode filtering. In order to compress 30 fs/5 mJ pulses to 5 fs/2.5 mJ pulses, in practical tests (S. Bohman et al., Opt. Express 16 (2008): p. 10684), a fiber with a diameter of 500 μm and having a length of 2.2 m was used. Additionally, a differentially pumped chamber (that was evacuated at the extremity where the pulses were coupled) had to be used in order to reduce nonlinear effects in front of the fiber, and therewith to reduce the energy coupled into the higher-order mode LP02 at the fiber entrance (input end). The differential-pumping scheme adds significant complexity to the setup, while the length of the HFC chamber alone was approximately 4 m, exceeding thus the size of a typical table-top setup.